Explicit Euler method

The Euler Polygonzugverfahren or explicit Euler method (also Euler-Cauchy method, or Euler-forward method) the simplest method for the numerical solution of an initial value problem .

It was presented by Leonhard Euler in his book Institutiones Calculi Integralis in 1768 . Cauchy used it to prove some uniqueness results for ordinary differential equations .

The procedure is sometimes referred to in physics as the small step method.

The procedure

Two steps of the explicit Euler method

For the numerical solution of the initial value problem:

{\ displaystyle {\ dot {y}} = f (t, y), \ quad \ quad y (t_ {0}) = y_ {0}}

for an ordinary differential equation choose a discretization step size , consider the discrete points in time ${\ displaystyle h> 0}$

{\ displaystyle t_ {k} = t_ {0} + kh, \ quad \ quad k = 0,1,2, \ dots}

and calculate the values

{\ displaystyle y_ {k + 1} = y_ {k} + hf (t_ {k}, y_ {k}), \ quad k = 0,1,2, \ dots}

The calculated values represent approximations to the actual values of the exact solution of the initial value problem. The smaller the selected step size , the more calculation work is required, but the more precise the approximated values become. ${\ displaystyle y_ {k}}$ ${\ displaystyle y (t_ {k})}$ ${\ displaystyle h}$

A modification of the method consists in choosing the step size to be variable. A sensible change in the step size requires an algorithm for step size control that estimates the error in the current step and then selects the step size for the next step accordingly.

If a method is defined via , the implicit Euler method is obtained . This is A-stable and therefore better suited for stiff initial value problems . ${\ displaystyle y_ {k + 1} = y_ {k} + hf (t_ {k + 1}, y_ {k + 1})}$

Derivation

For the derivation of one-step procedure which is an initial value problem mostly in the equivalent thereto integral equation reshaped

{\ displaystyle {\ begin {aligned} {\ dot {y}} & = f (t, y) \ ,, \ qquad y (t_ {0}) = y_ {0} \\\ Longleftrightarrow \ quad y (t ) & = y_ {0} + \ int _ {t_ {0}} ^ {t} f (s, y (s)) \, \ mathrm {d} s \,. \ end {aligned}}}

Now the idea is to use a simple quadrature formula for the integral in the explicit Euler method : the left-hand box rule . In every step one chooses the integrand as a constant value at the left limit of the interval ${\ displaystyle k}$

{\ displaystyle y (t_ {k + 1}) = y (t_ {k}) + \ int _ {t_ {k}} ^ {t_ {k + 1}} f (s, y (s)) \, \ mathrm {d} s \ approx y_ {k} + \ int _ {t_ {k}} ^ {t_ {k + 1}} f (t_ {k}, y_ {k}) \, \ mathrm {d} s = y_ {k} + hf (t_ {k}, y_ {k}) =: y_ {k + 1} \ ,.}

properties

Stability domain of the explicit Euler method

The explicit Euler method has consistency and convergence order 1. The stability function is and its stability area , therefore, the interior of the circle of radius 1 to -1 in the complex plane . ${\ displaystyle R (z) = 1 + z}$

Improved explicit Euler method

Instead of using the box rule for numerical integration, you can also use the midpoint rule.

{\ displaystyle \ int _ {t_ {k}} ^ {t_ {k + 1}} f (s, y (s)) \, \ mathrm {d} s \ approx hf \ left (t_ {k} + { \ frac {h} {2}}, y \ left (t_ {k} + {\ frac {h} {2}} \ right) \ right) \ ,.}

Now you turn back the explicit Euler method to approximate to ${\ displaystyle y \ left (t_ {k} + {\ frac {h} {2}} \ right)}$

{\ displaystyle y \ left (t_ {k} + {\ frac {h} {2}} \ right) \ approx y_ {k} + {\ frac {h} {2}} f (t_ {k}, y_ {k}) =: y_ {k + {\ frac {1} {2}}} \ ,.}

Together this leads to the improved explicit Euler method (or Euler method with a smaller step size )

{\ displaystyle y_ {k + 1} = y_ {k} + hf \ left (t_ {k} + {\ frac {h} {2}}, y_ {k + {\ frac {1} {2}}} \ right) \ quad {\ text {with}} \ quad y_ {k + {\ frac {1} {2}}} = y_ {k} + {\ frac {h} {2}} f (t_ {k}, y_ {k}) \ ,.}

Generalizations

It can essentially be generalized to more efficient processes by two different ideas.

The first idea is to include more than just one of the previously calculated values when calculating the next step. In this way, higher-order processes in the class of linear multistep processes are obtained .

The second idea is to evaluate the function on the interval at several points when calculating the next step . In this way the class of the Runge-Kutta methods is obtained . ${\ displaystyle f (t, y)}$ ${\ displaystyle [t_ {k}, t_ {k + 1}]}$

The class of general linear methods includes both ideas of generalization and contains the class of linear multi-step methods and the class of Runge-Kutta methods as a special case.

In addition, there is also an extension of the Euler method for stochastic differential equations : the Euler-Maruyama method .

literature

E. Hairer, SP Norsett, G. Wanner: Solving Ordinary Differential Equations I , Springer Verlag
M. Hermann: Numerics of ordinary differential equations, initial and boundary value problems , Oldenbourg Verlag, Munich and Vienna, 2004, ISBN 3-486-27606-9

Individual evidence

↑ Arnold Reusken: Numerical Analysis for Engineers and Scientists . Springer, Berlin 2006, ISBN 3-540-25544-3 , pp. 378 .
↑ Arnold Reusken: Numerical Analysis for Engineers and Scientists . Springer, Berlin 2006, ISBN 3-540-25544-3 , pp. 381 .
↑ Arnold Reusken: Numerical Analysis for Engineers and Scientists . Springer, Berlin 2006, ISBN 3-540-25544-3 , pp. 382 .

[1] Arnold Reusken: Numerical Analysis for Engineers and Scientists . Springer, Berlin 2006, ISBN 3-540-25544-3 , pp. 378 .

[2] Arnold Reusken: Numerical Analysis for Engineers and Scientists . Springer, Berlin 2006, ISBN 3-540-25544-3 , pp. 381 .

[3] Arnold Reusken: Numerical Analysis for Engineers and Scientists . Springer, Berlin 2006, ISBN 3-540-25544-3 , pp. 382 .